I am working on the reading from Athenaze 27b. I'm hoping someone can help me with a few points. My own humble attempt at translation is below the greek text.

Κῦρος δὲ αὐτίκα ἀπελαύνοντος Κροίσου μετὰ τὴν μάχην τὴν γενομενην ἐν τῇ Πτερίῃ,And Cyrus, as Croesus was at once marching away after the battle that happened in Pteria, (In this sentence, I put the underlined section together assuming that it forms a genitive phrase, was this correct?)

A few notes : "as soon as Croesus marched away" I would go for "knowing" for "επιστάμενος"I'm not sure what you translated as "certainly". If it's "μέλλοι" then its actual meaning is "going to", "plans to"As for your actual question "When he came to these conclusions", "When he decided so" or something like that would be a good translation I think. Sort of "the minute he decided to do so, he acted on it as fast as possible" kind of thing.

I couldn't figure out what the οἱ was supposed to be, and was embarrassed to ask. Now that you explained it, I just googled reflexive pronouns, and it is listed an "indirect reflexive pronoun". Maybe that's an advanced topic, because Athenaze certainly hasn't mentioned it.

So this phrase translated literally...

"since to him(self) these things seemed good, he also began to do them quickly..."

With the indication that the first part of the phrase indicates decision-making.